Tensor Algebra and Tensor Analysis for Engineers

1. Front Matter

   Pages i-xix

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2. Vectors and Tensors in a Finite-Dimensional Space

   • Mikhail Itskov

   Pages 1-36

3. Vector and Tensor Analysis in Euclidean Space

   • Mikhail Itskov

   Pages 37-72

4. Curves and Surfaces in Three-Dimensional Euclidean Space

   • Mikhail Itskov

   Pages 73-97

5. Eigenvalue Problem and Spectral Decomposition of Second-Order Tensors

   • Mikhail Itskov

   Pages 99-121

6. Fourth-Order Tensors

   • Mikhail Itskov

   Pages 123-135

7. Analysis of Tensor Functions

   • Mikhail Itskov

   Pages 137-171

8. Analytic Tensor Functions

   • Mikhail Itskov

   Pages 173-193

9. Applications to Continuum Mechanics

   • Mikhail Itskov

   Pages 195-215

10. Solutions

    • Mikhail Itskov

    Pages 217-290

11. Back Matter

    Pages 291-300

    PDF

This is the fifth and revised edition of a well-received textbook that aims at bridging the gap between the engineering course of tensor algebra on the one hand and the mathematical course of classical linear algebra on the other hand. In accordance with the contemporary way of scientific publication, a modern absolute tensor notation is preferred throughout. The book provides a comprehensible exposition of the fundamental mathematical concepts of tensor calculus and enriches the presented material with many illustrative examples. As such, this new edition also discusses such modern topics of solid mechanics as electro- and magnetoelasticity. In addition, the book also includes advanced chapters dealing with recent developments in the theory of isotropic and anisotropic tensor functions and their applications to continuum mechanics. Hence, this textbook addresses graduate students as well as scientists working in this field and in particular dealing with multi-physical problems. In each chapter numerous exercises are included, allowing for self-study and intense practice. Solutions to the exercises are also provided.

• Christoffel symbols
• Coordinate transformation
• Covariant and contravariant derivatives
• Derivative of the stretch and rotation tensor
• Generalized Rivlin’s Identities
• Polar decomposition of the deformation gradient
• Spectral decomposition of second-order tensors
• Tensor algebra
• Tensor-valued functions
• Tensors in finite-dimensional space
• matrix theory

• Department of Continuum Mechanics, RWTH Aachen University, Aachen, Germany

  Mikhail Itskov

Prof. Itskov studied Automobile Engineering at the Moscow State Automobile and Road Technical University, Russia. In 1990 he received his doctoral degree in mechanics, and in 2002 he obtained his habilitation degree in mechanics from the University of Bayreuth, Germany. Since 2004 he has been full professor for continuum mechanics at the RWTH  Aachen University, Germany. His research interests comprise tensor analysis, non-linear continuum mechanics, in particular the application to anisotropic materials, as well as the mechanics of elastomers and soft tissues in a broad sense.

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